The graph of any first-degree equation in two variables,
\begin{gather*}
Ax + By = C
\end{gather*}
is a line (as long as \(A\) and \(B\) are not both 0). A second-degree equation in two variables has the general form
\begin{gather*}
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
\end{gather*}
where \(A\text{,}\)\(B\text{,}\) and \(C\) cannot all be zero (because in that case the equation would not be second degree). The graphs of such equations are curves called conic sections because they are formed by the intersection of a plane and a cone, as illustrated below. Except for a few special cases called degenerate conics, the conic sections fall into four categories called circles, ellipses, hyperbolas, and parabolas.
Conic sections whose centers (or vertices, in the case of parabolas) are located at the origin are called central conics.
Subsection9.3.1Circles and Ellipses
The circle is the most familiar of the conic sections. Recall that the standard equation for a circle of radius, \(r\text{,}\) centered at the point \((h,k)\) is:
Notice that the denominators of both the \(x^2\)- and \(y^2\) -terms are \(r^2\text{.}\) You can check that the \(x\)- and \(y\)-intercepts of this circle are \((0, \pm r)\) and \((\pm r, 0)\text{.}\)
If the denominators of the \(x\)-squared and \(y\)-squared terms are not equal, the graph is called an ellipse. An ellipse is an elongated circle, or oval. Ellipses appear in a variety of applications. The orbits of the planets and of satellites about the earth are ellipses. The arches in some bridges are elliptical in shape, and whispering domes, such as the ceiling of the Mormon Tabernacle in Salt Lake City, are made from ellipses.
Recall that a circle is the set of all points in a plane that lie at a fixed distance from its center. An ellipse also has a geometric definition.
Definition9.3.1.Ellipse.
An ellipse is the set of points in the plane, the sum of whose distances from two fixed points (the foci) is a constant.
Using the distance formula and the definition above, we can show that the equation of an ellipse centered at the origin has the following standard form.
Central Ellipse.
The equation of an ellipse centered at the origin is
By setting \(y\) equal to zero in the equation above, we find that the \(x\)-intercepts of this ellipse are \(a\) and \(-a\text{;}\) by setting \(x\) equal to zero, we find that the \(y\)-intercepts are \(b\) and \(-b\text{.}\)
The line segment that passes through the foci (labeled \(F_1\) and \(F_2\) on the graphs below) and ends on the ellipse is called the major axis. If \(a \gt b\text{,}\) the major axis is horizontal, as shown in the Figure below left. The \(x\)-intercepts are the endpoints of the major axis, so its length is \(2a\text{.}\) The vertical segment with length \(2b\) is called the minor axis. The endpoints of the major axis are the vertices of the ellipse and the endpoints of the minor axis are the covertices.
If \(a \lt b\text{,}\) the major axis is vertical and has length \(2b\text{.}\) In this case the endpoints of the major axis are the \(y\)-intercepts of the ellipse. (See Figure above right.) The minor axis is horizontal and has length \(2a\text{.}\)
The standard form of the equation for an ellipse gives us enough information to sketch its graph.
The graph is an ellipse with major axis on the \(y\)-axis. Because \(a^2 = 8\) and \(b^2 = 25\text{,}\) the vertices are located at \((0, 5)\) and \((0, -5)\text{,}\) and the covertices lie \(\sqrt{8} \) units to the right and left of the center, or approximately at \((2.8, 0)\) and \((-2.8, 0)\text{.}\)
To sketch the ellipse, we first locate the vertices and covertices. Then we draw a smooth curve through the points. The graph of \(\dfrac{x^2}{8}+\dfrac{y^2}{25}=1 \) is shown below.
The equation of any central ellipse may be written as
\begin{gather*}
Ax^2+By^2=C
\end{gather*}
where \(A\text{,}\)\(B\text{,}\) and \(C\) the same sign. The features of the graph are easier to identify if we first convert the equation to standard form.
Because \(a^2=3\) and \(b^2=12\text{,}\) the vertices are \((0,\pm 2\sqrt{3}) \) and the covertices are \((\pm \sqrt{3},0) \text{.}\) We plot points at about \((0, \pm 3.5)\) and \((\pm 1.7, 0)\text{,}\) then draw an ellipse through the points, as shown at right.
Find the exact coordinates of any points with \(y\)-coordinate 2 on the ellipse \(4x^2 + y^2 = 12\text{.}\) Plot and label those points on the ellipse.
Solve the equation \(4x^2 + y^2 = 12\) when \(y = -4\text{.}\) What do the solutions tell you about the graph of the ellipse?
The horizontal axis of the ellipse has length \(2a\text{,}\) and the vertical axis has length \(2b\text{,}\) the same as for central ellipses. When \(a \gt b\text{,}\) the major axis is horizontal and the ellipse is short and wide. When \(a \lt b\text{,}\) the major axis is vertical and the ellipse is tall and narrow, as shown below.
The graph is an ellipse with center at \((-2,1)\text{.}\) We have \(a=4\) and \(b=\sqrt{5}\text{,}\) and the major axis is parallel to the \(x\)-axis because \(a \gt b\text{.}\) We plot the vertices four units to the left and right of the center, at \((-6,1)\) and \((2,1)\text{.}\) The covertices lie \(\sqrt{5}\) units above and below the center, at approximately \((-2, 3.2)\) and \((-2, -1.2)\text{.}\) The graph is shown below.
We set \(y=\alert{0}\) and solve the resulting equation to find the \(x\)-intercepts.
The \(x\)-intercepts are \(\left(-2 \pm \dfrac{8\sqrt{5}}{5}, 0\right)\) or approximately \((1.6,0)\) and \((-5.6,0)\text{.}\) We set \(x=\alert{0}\) to find the \(y\)-intercepts.
Second-degree equations in which the coefficients of \(x^2\) and \(y^2\) have the same sign can be written in one of the standard forms for an ellipse by completing the square. The equation can be graphed easily from the standard form.
We complete the square in \(x\) by adding \(\alert{4}\) to \(x^2-4x\text{,}\) and adding \(4 \cdot \alert{4}\text{,}\) or \(\blert{16}\text{,}\) to the right side of the equation. We complete the square in \(y\) by adding \(\alert{1}\) to \(y^2-2y\text{,}\) and \(9 \cdot \alert{1}\text{,}\) or \(\blert{9}\text{,}\) to the right side.
We write each term on the left side as a perfect square to get
\begin{align*}
4(x-2)^2 + 9(y-1)^2 \amp= 36 \amp\amp \blert{\text{Divide both sides by 36.}} \\
\frac{(x-2)^2}{9} + \frac{(y-1)^2}{4} = 1
\end{align*}
The graph is an ellipse with center at \((2, 1),~a^2=9\text{,}\) and \(b^2=4\) The vertices lie 3 units to the right and left of the center at \((5,1)\) and \((-1,1)\text{;}\) the covertices lie 2 units above and below the center at \((2,3)\) and \((2,-1)\text{.}\) The graph is shown below.
Caution9.3.13.
When completing the square in the Example above, do not forget the coefficients you factored out in the first step. When we add 4 to complete the square in \(x\text{,}\) it is multiplied by a factor of 4, so we must add \(4 \cdot 4\) or 16 to the right side of the equation. Similarly, we must add \(9 \cdot 1\) or 9 to the right side when we complete the square in \(x\text{.}\)
To write the equation of an ellipse from a description of its properties, we must find the center of the ellipse and the lengths of its axes. We can then substitute this information into the standard form.
You may find it helpful to plot the given points to help you visualize the ellipse. The center of the ellipse is the midpoint of the major (or minor) axis.
\begin{align*}
h \amp = \overline{x} = \dfrac{1+5}{2} = 3\\
k \amp = \overline{y} = \dfrac{3-5}{2} = -1
\end{align*}
Thus, the center is the point \((3, -1)\text{.}\) The horizontal axis is shorter, and \(a\) is the distance between the center and either covertex, say \((5, −1)\) Thus,
\begin{gather*}
a = 5-3 = 2
\end{gather*}
The value of \(b\) is the distance from the center to one of the vertices, say \((3,3)\text{:}\)
A doorway is topped by a semi-elliptical arch. The doorway is 230 centimeters high at its highest point and 200 centimeters high at its lowest point. It is 80 centimeters wide.
Find an equation for the ellipse.
How high is the doorway 8 centimeters from the left side?
31.
The wing of a World War II British Spitfire is an ellipse whose major axis is 48 feet. The minor axis is 16 feet, but part of the ellipse is cut off parallel to the major axis. This cut edge is 46 feet long.
Find an equation for the ellipse.
How wide is the wing at its center? Round your answer to two decimal places.
The centerline of a sailboat from bow to stern along the bottom (its keel) is elliptical in shape, with a major axis of 360 centimeters. The minor axis of the ellipse is 100 centimeters, but the deck of the sailboat (the top of the ellipse) has been cut off parallel to the major axis. The deck of the sailboat is 330 centimeters long.
Find an equation for the ellipse.
What is the maximum distance from the deck to the bottom of the keel? Round your answer to two decimal places.